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INSTRUCTOR’S MANUAL FOR COMPUTABILITY AND LOGIC FIFTH EDITION PART A. FOR ALL READERS JOHN P. BURGESS Professor of Philosophy Princeton University jburgess@princeton.edu Note This work is subject to copyright, but instructors who adopt Computability & Logic as a textbook are hereby authorized to copy and distribute the present Part A This permission does not extend to Part B Contents Dependence of Chapters (Leitfaden) General Remarks on Problems (for Students) Hints for OddNumbered Problems Computability Theory Hints for OddNumbered Problems Basic Metalogic 11 Errata 20 Dependence of Chapters General Remarks on Problems (for Students) • The problems for each chapter should be read as part of that chapter, even those that are not assigned. They often add important information not covered in the text • The results of earlier problems, whether or not assigned, may be used in later problems. Many problems are parts of sequences • Before working the problems for any chapter, check to see whether there are any errata listed for that chapter, and especially for its problems • Hints are provided for oddnumbered problems in chapters 118. The hints for some problems are inevitably more substantial than those for others Hints for OddNumbered Problems: Computability Theory (Chapters 18) Chapter 1 1.1 The converse assertion then follows from the first assertion by applying it to f 1 and its inverse f 11 1.3 For (a) consider the identity function i(a) = a for all a in A. For (b) and (c) use the preceding two problems, as per the general hint above 1.5 Show both sets are denumerable 1.7 If we can fix for each i an enumeration of Ai Ai = {ai1, ai2, ai3, … } Then we can enumerate A, which is the set of all aij for all i and j in the same way we enumerated pairs (i, j) in Example 1.2. However, when we assume that for each Ai there exists an enumeration of it, it follows that there exist many different such enumerations for each Ai; and when set theory is developed rigously, in order to conclude that there is a way of fixing simultaneously for each i some one, specific enumeration out of all the many different enumerations that exist, we need a principle known as the axiom of choice. As this is not a textbook of set theory, we are not going to go into such subtleties Chapter 2 2.1 Imitate the proof for the set of positive integers 2.3 You do not need to use trigonometry or give an analytical formula for the correspondence to do this problem; a simple geometric description of a correspondence will be enough 2.5 While this can be done using the preceding two problems, as per the general hint, for students who remember trigonometry, a correspondence can also be defined directly using the tangent function 2.7 Note that rational numbers whose denominator (when written in lowest terms) is a power of two have two binary representations, one ending in all 0’s and the other in all 1’s from some point on (as in 1/2 = .1000000… = .0111111…), while in every other case the binary representation is unique and does not involve all 0’s or all 1’s from any point on. 2.9 In addition to the immediately preceding problems, Problem 1.6 may be useful 2.11 Read carefully through the sequence of preceding problems 2.13 This is a philosophical rather than a mathematical question, and as such does not have a universally agreed answer, though there is a consensus that somehow defining a set in terms of the notion of definability itself is somehow to blame for the paradox Chapters 3 & 4 3.1 One state will be required in (a), two in (b) 3.3 Proceed as in Problem 3.1(b) but when reaching a blank in state 2 print a stroke and go into state 3. At this stage you will have a block of n strokes followed by a blank followed by a block of m + 1 + k strokes. In state 3 on a stroke move right and go into state 4. In state 4 on a stroke erase it. You will now have blocks of n, m + 1, and k 1 strokes. Take it from there 3.5 Proceed in cycles, during each of which you erase the leftmost stroke of the first block and the rightmost stroke of the second block, and add a stroke to a third block to the right of them both. When one of the two original blocks has been completely erased, erase also the other. The trick is to keep track of when this happens 4.1 It is certainly not possible just exploring without marking the tape 4.3 It is not possible to preserve the original block unaltered while making a copy 4.5 A description of a function of the kind a universal machine would have to compute is implicit in the discussion of the diagonal function in the text Chapter 5 5.1 Subtraction is to the predecessor function as addition is to the successor function 5.3 Use problem 5.1 5.5 Keep subtracting y from x, while checking each time you do so that what is left is still ≥ y 5.7 Manœuvres of just this kind take place the simulation of abacus machines by Turing machines 5.9 See preceding problems 5.11 See the proof of Theorem 4.1 Chapter 6 6.1 For instance, in (a), g(x, y) = f(id2, id2) 6.3 These can be done ‘from scratch’ or, generally more easily, by showing the indicated functions are compositions of functions already known to be primitive recursive 6.5 Proposition 6.5 may be useful 6.7 Each recursive function is denoted by some expression built up using Cn, Pr, and Mn from names for the zero, successor, and identity functions 6.9 Use the following fact: There is a recursive function f such that f(0) = 0 but f(x) is undefined for x > 0. (For instance, f(x) = the least y such that |x y| + y = 0.) 10 Chapter 7 7.1 Compare with Problem 6.1 7.3 Use Corollary 7.8 7.5 Consider the auxiliary function g(n) = the least element of A that is > n 7.7 Apply the preceding two problems to obtain a recursive function a and use it and the original f to define a suitable g 7.9 First show that the auxiliary function g(n) = J(f(n), f(n + 1)) is primitive recursive, where J is as in Problems 6.2 and 6.5 7.11 First introduce a suitable auxiliary function, as in Example 7.20 7.13 Suppose that ci and d are the numbers associated with gi and f respectively, so that gi(x1, … , xn)